Efficient Treatment of the Vardy-Brown Unsteady Shear in Pipe Transients

An accurate, simple, and efficient approximation to the Vardy-Brown unsteady friction equation is derived and shown to be easily implemented within a one-dimensional characteristics solution for unsteady pipe flow. For comparison, the exact Vardy-Brown unsteady friction equation is used to model shear stresses in transient turbulent pipe flows and the resulting waterhammer equations are solved by the method of characteristics. The approximate Vardy-Brown model is more computationally efficient (i.e., requires one-sixth the execution time and much less memory storage) than the exact Vardy-Brown model. Both models are compared with measured data from different research groups and with numerical data produced by a two-dimensional turbulence waterhammer model. The results show that the exact Vardy-Brown model and the approximate Vardy-Brown model are in good agreement with both laboratory and numerical experiments over a wide range of Reynolds number and wave frequencies. The proposed approximate model only requires the storage of flow variables from a single time step while the exact Vardy-Brown model requires the storage of flow variables at all previous time steps and the two-dimensional model requires the storage of flow variables at all radial nodes.